Ok. I don't know what that tab is connected to and the data sheet does not say, so it may not be connected to any of the MOSFET pins, electrically, just thermally to the mosfet's chip. So connecting it to anything else is not intended by the manufacturers. Given the gate is floating, touching it with a finger could cause a random change, so the effects you saw don't mean anything important. Try that 10K resistor between the gate and 5V.